On a Branch and Bound Algorithm for the Solution of a Family of Softening Material Problems of Mechanics with Applications to the Analysis of Metallic Structures

Abstract

The variational formulation of mechanical problems involving nonmonotone,possibly multivalued, material or boundary laws leads to hemivariationalinequalities. The solutions of the hemivariational inequalities constitutesubstationarity points of the related energy (super)potentials. For theircomputation convex and global optimization algorithms have been proposedinstead of the earlier nonlinear optimization methods, due to the lack ofsmoothness and convexity of the potential. In earlier works one of us hasproposed an approach based on the decomposition of the solutions space intoconvex parts resulting in a sequence of convex optimization subproblems withdifferent feasible sets. In this case nonconvexity of the potential wasattributed to (generalized) gradient jumps. In order to treat ’softening‘material effects, in the present paper this method is extended to cover alsoenergy functionals where nonconvexity is caused by the existence of concavesections. The nonconvex minimization problem is formulated as d.c.(difference convex) minimization and an algorithm of the branch and boundtype based on simplex partitions is adapted for its treatment. Thepartitioning scheme employed here is adapted to the large dimension of theproblem and the approximation steps are equivalent to convex minimizationsubproblems of the same structure as the ones arising in unilateral problemsof mechanics. The paper concludes with a numerical example and a discussionof the properties and the applicability of the method.